In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a simplicial complex is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
composed of
points,
line segments,
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s, and their
''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a
simplicial set appearing in modern simplicial
homotopy theory. The purely
combinatorial counterpart to a simplicial complex is an
abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.''
[, Section 4.3]''
Definitions
A simplicial complex
is a set of
simplices that satisfies the following conditions:
:1. Every
face of a simplex from
is also in
.
:2. The non-empty
intersection of any two simplices
is a face of both
and
.
See also the definition of an
abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
A simplicial ''k''-complex
is a simplicial complex where the largest dimension of any simplex in
equals ''k''. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
or higher-dimensional simplices.
A pure or homogeneous simplicial ''k''-complex
is a simplicial complex where every simplex of dimension less than ''k'' is a face of some simplex
of dimension exactly ''k''. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a ''non''-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as
triangulations and provide a definition of
polytopes.
A facet is a maximal simplex, i.e., any simplex in a complex that is ''not'' a face of any larger simplex.
[.] (Note the difference from a
"face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of)
simplicial polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz ...
s this coincides with the meaning from polyhedral combinatorics.
Sometimes the term ''face'' is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
For a simplicial complex
embedded in a ''k''-dimensional space, the ''k''-faces are sometimes referred to as its cells. The term ''cell'' is sometimes used in a broader sense to denote a set
homeomorphic to a simplex, leading to the definition of
cell complex.
The underlying space, sometimes called the carrier of a simplicial complex is the
union of its simplices. It is usually denoted by
or
.
Support
The
relative interior
In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.
Formally, the relative interior of a set S (den ...
s of all simplices in
form a partition of its underlying space
: for each point
, there is exactly one simplex in
containing
in its relative interior. This simplex is called the support of ''x'' and denoted
.''
[, Section 4.3]''
Closure, star, and link
File:Simplicial complex closure.svg, Two and their .
File:Simplicial complex star.svg, A and its .
File:Simplicial complex link.svg, A and its .
Let ''K'' be a simplicial complex and let ''S'' be a collection of simplices in ''K''.
The closure of ''S'' (denoted
) is the smallest simplicial subcomplex of ''K'' that contains each simplex in ''S''.
is obtained by repeatedly adding to ''S'' each face of every simplex in ''S''.
The
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
of ''S'' (denoted
) is the union of the stars of each simplex in ''S''. For a single simplex ''s'', the star of ''s'' is the set of simplices having ''s'' as a face. The star of ''S'' is generally not a simplicial complex itself, so some authors define the closed star of S (denoted
) as
the closure of the star of S.
The
link of ''S'' (denoted
) equals
. It is the closed star of ''S'' minus the stars of all faces of ''S''.
Algebraic topology
In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, simplicial complexes are often useful for concrete calculations. For the definition of
homology groups of a simplicial complex, one can read the corresponding
chain complex directly, provided that consistent orientations are made of all simplices. The requirements of
homotopy theory lead to the use of more general spaces, the
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at
Polytope of simplicial complexes as subspaces of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
made up of subsets, each of which is a
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
. That somewhat more concrete concept is there attributed to
Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
(see , , ).
Combinatorics
Combinatorialists often study the ''f''-vector of a simplicial d-complex Δ, which is the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
sequence
, where ''f''
''i'' is the number of (''i''−1)-dimensional faces of Δ (by convention, ''f''
0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the
octahedron, then its ''f''-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its ''f''-vector is (1, 18, 23, 8, 1). A complete characterization of the possible ''f''-vectors of simplicial complexes is given by the
Kruskal–Katona theorem.
By using the ''f''-vector of a simplicial ''d''-complex Δ as coefficients of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
(written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the ''f''-polynomials would be
and
, respectively.
Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging ''x'' − 1 into the ''f''-polynomial of Δ. Formally, if we write ''F''
Δ(''x'') to mean the ''f''-polynomial of Δ, then the h-polynomial of Δ is
:
and the ''h''-vector of Δ is
:
We calculate the h-vector of the octahedron boundary (our first example) as follows:
:
So the ''h''-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this ''h''-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial
polytope (these are the
Dehn–Sommerville equations In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their ge ...
). In general, however, the ''h''-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting ''h''-vector is (1, 3, −2).
A complete characterization of all simplicial polytope ''h''-vectors is given by the celebrated
g-theorem In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions ...
of
Stanley, Billera, and Lee.
Simplicial complexes can be seen to have the same geometric structure as the
contact graph
In the mathematical area of graph theory, a contact graph or tangency graph is a graph whose vertices are represented by geometric objects (e.g. curves, line segments, or polygons), and whose edges correspond to two objects touching (but not c ...
of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of
sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...
s, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
Computational problems
The
simplicial complex recognition problem is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is
undecidable for any ''d''-dimensional manifolds for ''d'' ≥ 5.
See also
*
Abstract simplicial complex
*
Barycentric subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...
*
Causal dynamical triangulation
*
Delta set
In mathematics, a Δ-set ''S'', often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. ...
*
Polygonal chain 1 dimensional simplicial complex
*
Tucker's lemma
References
*
*
*
External links
*
Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.
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Topological spaces
Algebraic topology
Simplicial sets
Triangulation (geometry)